On the comparison of the density type topologies generated by sequences and by functions

Małgorzata Filipczak, Tomasz Filipczak

On the comparison of the density type topologies generated by sequences and by functions

Abstract. In the paper we investigate density type topologies generated by functions f satisfying condition lim inf

x→0⁺ f(x)

x

> 0, which are not generated by any sequence.

2000 Mathematics Subject Classification: 54A10, 26A15, 28A05.

Key words and phrases: density points, density topology, comparison of topologies.

Through the paper we shall use standard notation: ℝ will be the set of real numbers, ℕ the set of positive integers, L the family of Lebesgue measurable subsets of ℝ and |E| the Lebesgue measure of a measurable set E. By Φ ^d (E) we shall denote the set of all Lebesgue density points of measurable set E (i.e. Φ d (E) = n x ∈ ℝ; lim h →0

|(x−h;x+h)∩E|

2h = 1 o

) and by T ^d the density topology consists of measurable sets satisfying E ⊂ Φ ^d (E). For any operators Φ, Ψ : L → L we write Φ ⊂ Ψ if Φ (E) ⊂ Ψ (E) for every E ∈ L. If Φ ⊂ Ψ and Φ 6= Ψ then we write Φ ⊈ Ψ.

We will consider two generalizations of Lebesgue density. First of them, called a density generated by a sequence, was introduced by J. Hejduk and M. Filipczak in [6]. For a convenience we will formulate definitions using decreasing sequences tending to zero, instead of nondecreasing sequences going to infinity.

Let e S be the family of all decreasing sequences tending to zero. We will denote sequences from e S by (a ⁿ ) or by hai. Let hai ∈ e S, E ∈ L and x ∈ ℝ. We shall say that x is an hai-density point of E (a right-hand hai-density point of E) if

n lim →∞

|E ∩ [x − a ⁿ ; x + a n ]|

2a n

= 1 ( lim

n →∞

|E ∩ [x; x + a ⁿ ]|

a n

= 1).

By Φ _hai (E) (Φ ⁺ _hai (E)) we will denote the set of all hai-density (right-hand hai-

density) points of E. In the same way one may define left-hand hai-density point of

E and the set Φ ⁻ _hai (E). Evidently, Φ _hai (E) = Φ ⁺ _hai (E) ∩ Φ ⁻ _hai (E).

In [6] it was proved that Φ _hai is a lower density operator and the family T hai = 

E ∈ L; E ⊂ Φ hai (E)

is a topology containing the density topology T ^d . Moreover for a n = _n ¹ , Φ _hai = Φ d

and T hai = T ^d .

For any pair of sequences hai and hbi from e S we denote by hai∪hbi the decreasing sequence consisting of all elements from hai and hbi. It is clear that hai ∪ hbi ∈ e S and that

Proposition 1 If Φ _hai ⊂ Φ hbi then Φ _hai∪hbi = Φ _hai .

The second type of density we will observe are densities generated by functions.

We denote by A the family of all functions f : (0; ∞) → (0; ∞) such that (A1) lim

x→0

f (x) = 0, (A2) lim inf

x →0

f(x)

x < ∞, (A3) f is nondecreasing.

Let f ∈ A. We say that x is a right-hand f-density point of a measurable set E if

h→0 lim

|(x; x + h) \ E|

f (h) = 0.

By Φ ⁺ _f (E) we denote the set of all right-hand f-density points of E. In the same way one may define left-hand f-density points of E and the set Φ ⁻ _f (E). We say that x is an f -density point of E if it is a right and a left-hand f-density point of E. By Φ f (E) we denote the set of all f-density points of E, i.e. Φ f (E) = Φ ⁺ _f (E) ∩ Φ ⁻ f (E).

For any f ∈ A, the family

T ^f = {E ∈ L; E ⊂ Φ ^f (E)}

forms a topology stronger than the natural topology on the real line (see [1, Th. 7]

and [4, Th. 1]).

Proposition 2 ([4, Prop. 4]) For each f, g ∈ A, an inclusion T ^f ⊂ T ^g holds if and only if Φ f ⊂ Φ ^g .

In [2] and [3] it has been shown that properties of f-density operator Φ f and f -density topology T ^f are strictly depended on lim inf

x →0

f(x)

x . In our paper we are interested in topologies generated by functions f ∈ A for which lim inf

x→0

f(x)

x > 0.

The family of all such functions we will denote by A ¹ .

Topologies generated by functions from A ¹ and from A \ A ¹ have quite different

properties (for example they satisfy different separation axioms - see [3, Th. 5 and

Th. 7)]). On the other hand for any function f from A ¹ properties of T ^f are similar to properties of topologies generated by sequences, so similar to properties of the density topology T ^d (compare [4, Th. 3], [9], [7] and [8]). Moreover

Proposition 3 ([4, Th. 5]) For any sequence hsi ∈ e S, the function f defined by a formula

f (x) = s n for x ∈ (s ⁿ⁺¹ ; s n ] belongs to A ¹ and Φ _hsi = Φ f .

Corollary 4 For each hai , hbi ∈ e S, an inclusion T hai ⊂ T hbi holds if and only if Φ _hai ⊂ Φ hbi .

In [4] we have constructed a function f ∈ A ¹ such that Φ f 6= Φ hsi for each hsi ∈ e S. We will remain the definition of that function, omitting the proof, because we will construct a similar one in Theorem 11.

Example 5 ([4, Th. 6]) Let us define sequences hwi = (2, 2, 3, 3, 3, 4, 4, 4, 4, . . .) ,

hri = (1, 2, 1, 2, 3, 1, 2, 3, 4, . . .) , a 0 = 1, a n = a n −1

w ² _n for n ­ 1, b n = a n w n = a n −1

w n for n ­ 1.

Evidently

n lim →∞

a n −1

b n

= lim

n →∞

b n

a n = ∞.

The function f defined by a formula f (x) =

 a n −1 for x ∈ (b ⁿ ; a n −1 ] , b n r n for x ∈ (a ⁿ ; b n ] . belongs to A ¹ and Φ f 6= Φ hsi for each hsi ∈ e S.

The function constructed above proves that the family 

T ^f ; f ∈ A ¹ is big- ger than n

T hsi ; hsi ∈ e S o

. We will show that there exist continuum topologies from

 T ^f ; f ∈ A ¹

\ n

T hsi ; hsi ∈ e S o

and that for any pair of sequences hai , hbi ∈ e S satis- fying T hai ⊈ T hbi there is a function f ∈ A ¹ such that T hai ⊈ T ^f ⊈ T hbi and T ^f 6= T hsi

for hsi ∈ e S.

Set

f α (x) = f  x α

 and s α (n) = αs (n)

for f ∈ A ¹ , hsi ∈ e S and α > 0. Obviously f ^α ∈ A ¹ and hs ^α i ∈ e S.

Proposition 6 If Φ f = Φ _hsi then Φ f

= Φ _hs

_i .

Proof It is sufficient to prove that for any measurable set E 0 ∈ Φ ⁺ f

(E) ⇐⇒ 0 ∈ Φ ⁺ _hs

i (E) . If 0 ∈ Φ ⁺ f

(E) then

 (0; x) ∩ _α ¹ E ⁰  

f (x) =

1

α |(0; αx) ∩ E ⁰ | f α (αx)

x →0+

−→ 0,

and so 0 ∈ Φ ⁺ f 1

α E
= Φ ⁺ _{hsi α} ¹ E
. Hence

|(0; s ^α (n)) ∩ E ⁰ | s α (n) =

 (0; s (n)) ∩ _α ¹ E ⁰   s (n)

n →∞

−→ 0,

which gives 0 ∈ Φ ⁺ _hs

_i (E).

Conversely, if 0 ∈ Φ ⁺ _hs

i (E) then  (0; s (n)) ∩ α ¹ E ⁰  

s (n) = | (0; s α (n)) ∩ E ⁰ | s α (n)

n→∞ −→ 0,

and consequently 0 ∈ Φ ⁺ _hsi α ¹ E
= Φ ⁺ _{f α} ¹ E
. Thus

|(0; x) ∩ E ⁰ |

f α (x) = α   0; _α ^x

∩ α ¹ E ⁰  

f ^x _α
^x→0+ −→ 0,

which implies 0 ∈ Φ ⁺ f

(E). _■

Corollary 7 If Φ f ∈ / n

Φ _hsi ; hsi ∈ e S o

then Φ f

∈ / n

Φ _hsi ; hsi ∈ e S o

for any α > 0.

Theorem 8 If f is the function defined in Example 5 then Φ f

⊈ Φ ^f

for any α > β > 1.

Proof Since f is nondecreasing, we have f α ¬ f ^β and Φ f

⊂ Φ ^f

. Let (n i ) be an increasing sequence of positive integers such that r n

= 1 for every i. Define

E = ℝ \ [ ∞ i=1

βb n

; βb n

√ w n

.

It is sufficient to show that

0 ∈ Φ ^f

(E) \ Φ ^f

(E) .

Without loss of generality, we can assume that for any n βa n < b n and βb n < αb n < βb n √ w n < a n −1 . If x ∈ (βb ⁿ

; βa n

−1 ] for some i then

(1) |(0; x) ∩ E ⁰ |

f β (x) ¬ βb n

√ w n

f 

x β

 = βb n

√ w n

a n

−1

= β

√ w n

.

If x ∈ (βa ⁿ

; βb n

] for some i then

(2) |(0; x) ∩ E ⁰ |

f β (x) < a n

f 

x β

 = a n

r n

b n

= 1 w n

.

If x ∈ (βa ^p ; βb p −1 ] and n i < p < n i+1 for some i then

(3) |(0; x) ∩ E ⁰ |

f β (x) < a p

r p b p ¬ 1 w p

. From (1)-(3) we conclude that 0 ∈ Φ ^f

(E).

On the other hand for x i = αb n

|(0; x ⁱ ) ∩ E ⁰ |

f α (x i ) ­ |(βb ⁿ

; αb n

)|

f (b n

) = (α − β) b ⁿ

r n

b n

= α − β > 0,

and consequently 0 / ∈ Φ ^f

(E). _■

In [4, Th. 1] it has been proved that for each function f ∈ A there is a continuous function g ∈ A such that Φ ^f = Φ g . Thus there is at most continuum different f- density topologies. Combining this result with Corollary 4, Corollary 7 and Theorem 8 we obtain

Corollary 9 The family 

T ^f ; f ∈ A ¹

\ n

T hsi ; hsi ∈ e S o

has cardinality continuum.

In the next theorem we will compare densities generated by sequences. Let us formulate a useful lemma from [5]. Fix hai and hbi from e S. There is an unique sequence (k n ) of positive integers such that

b n ∈ (a ^k

+1 ; a k

]

for each n with b n < a 1 . From now on (k n ) will denote this unique sequence.

Lemma 10 ([5, Th. 7]). The following conditions are equivalent (a) Φ _hai ⊂ Φ hbi .

(b) For arbitrary increasing sequence (n i ) of positive integers lim inf

i→∞

a k

b n

< ∞ or lim inf _i→∞ b n

a k

+1 = 1.

Theorem 11 Let hai , hbi ∈ e S. If Φ hbi ⊈ Φ hai then there is f ∈ A ¹ such that Φ _hbi ⊂ Φ ^f ⊂ Φ hai and Φ f 6= Φ hsi for hsi ∈ e S.

Proof By Proposition 1 we can asume that hai is a subsequence of hbi. Since Φ _hai ⊈ Φ hbi , there exists an increasing sequence (n i ) of positive integers such that

i lim →∞

a k

b n

= ∞ and M = lim inf _i

→∞

b n

a k

+1 > 1.

Let us denote

β =

 √

M ; M < ∞

2 ; M = ∞ .

Replacing (n _i ) by a subsequence we can assume that (k _n

) is increasing and for every i

(4) ^a _b

ni

> i ² , _a ^b

kni +1

> β ³ and _a ^b

kni +1

< β ⁵ if M < ∞.

Let us define

hci = hai ∪ (b ⁿ

) ,

hri = (1, 2, 1, 2, 3, 1, 2, 3, 4, . . .) , g 1 (x) = a n for x ∈ (a ⁿ⁺¹ ; a n ] , g 2 (x) = c n for x ∈ (c ⁿ⁺¹ ; c n ] , g 3 (x) = b n for x ∈ (b ⁿ⁺¹ ; b n ] ,

f (x) =

 r i b n

for x ∈ a ^k

⁺¹ ; b n

 , g 1 (x) for x / ∈ S ∞

i=1 a k

+1 ; b n

 .

Since 1 ¬ r i ¬ i ² , f ∈ A ¹ and g ₃ ¬ g ² ¬ f ¬ g ¹ . Thus

Φ _hbi = Φ g

⊂ Φ hci = Φ g

⊂ Φ ^f ⊂ Φ ^g

= Φ _hai .

We will show that Φ f 6= Φ hsi for hsi ∈ e S. Suppose, contrary to our claim, that there is hsi ∈ e S such that

(5) Φ f = Φ _hsi .

Let us consider the sets

T = 

i ∈ ℕ; ∃ ^m ∈ℕ s m ∈ 

β ³ a k

+1 ; ib n

 , R = {r ⁱ ; i ∈ T } .

We will prove that the set R is bounded. Suppose it is not true and so T is infinite.

Let (t i ) be an increasing sequence consisting of all elements from T , i.e.

T = {t ⁱ ; i ∈ ℕ} .

Let m i be a fixed positive integer such that s m

∈ h

β ³ a k

+1 ; t i b n

i .

We will show that

(6) s m

< 2b n

for almost every i. Suppose on the contrary that there is an increasing sequence (p i ) of positive integers such s m

­ 2b ⁿ

for every i. Hence

a k

i

s m

­ t ² _p

b n

t p

b n

= t p

­ p ⁱ ­ i

and so

i lim →∞

a k

i

s

= ∞ and

lim inf

i →∞

s

b n

­ 2.

This, by Lemma 10, contradicts the assumption that Φ _hci ⊂ Φ hsi and finishes the proof of (6).

Now we show that there are real numbers ε > δ > 0 such that (7) a k

+1 < δb n

< εb n

< s m

for almost every i. If M = lim inf

i →∞

b

a

< ∞ then by (4) it is sufficient to set ε = _β ¹

and δ = _β ¹

. If M = ∞, then obviously it is sufficient to show that for some positive ε inequality

εb n

< s m

holds for almost every i. Suppose on the contrary that there is an increasing sequence (p i ) of positive integers such is m

< b n

for every i. Hence

i lim →∞

b n

s m

= ∞.

and

lim inf

i →∞

s m

a k

i

+1 ­ β ³ > 1.

which also contradicts the assumption that Φ _hci ⊂ Φ hsi and ends the proof of (7).

By our assumptions, the set R is not bounded. Thus there exists an increasing sequence (p i ) of positive integers such that

i→∞ lim r t

= ∞.

Put

A = [ ∞ i=1

 δb n

; εb n

 .

From (6) and (7) it follows that  A ∩ 0; s m



s m

­ εb n

− δb ⁿ

2b n

= ε − δ 2 > 0 for almost every i, and so

(8) 0 / ∈ Φ ⁺ _hsi (A ⁰ ).

We will show that 0 ∈ Φ ⁺ f (A ⁰ ). Let x ∈ (a ^j ; a j −1 ]. If j = k n

for some i then

(9) |A ∩ (0; x)|

f (x) ¬ εb n

r t

b n

= ε r t

.

On the other hand if k m

< j < k m

for some i then

(10) |A ∩ (0; x)|

f (x) ¬ εb n

a j+1 ¬ εb n

a k

i+1

¬ ε

t ² _p

¬ ε (i + 1) ² .

From (5), (9) and (10) it follows that 0 ∈ Φ ⁺ f (A ⁰ ) = Φ ⁺ _hsi (A ⁰ ), contrary to (8). Hence we conclude that R is bounded.

Let L be a positive integer such that L > sup R,

(p i ) be an increasing sequence of positive integers for which r p

= L

and

B = [ ∞ i=1

 1

β b n

; b n

 .

Since 

B ∩ 0; b ⁿ



f (b n

) ­ b n

− β ¹ b n

r p

b n

= 1 − ¹ β

L > 0,

(11) 0 / ∈ Φ ⁺ f (B ⁰ ).

We will prove that 0 ∈ Φ ⁺ _hsi (B ⁰ ). By definition of (p i ), p i ∈ T for every i and / consequently

s m ∈ / h

β ³ a k

+1 ; p i b n

i

for m, i ∈ ℕ. Let us consider any term s ^m of hsi. There is a positive integer j such that s m ∈ (a ^j ; a _j−1 ]. If j = k n

for some i then

s m < β ³ a k

+1 or s m > p i b n

. In the first case we have

(12) |B ∩ (0; s ^m )|

s m ¬ b n

a k

+1 ¬ b n

a k

¬ 1

p ² _i+1 ¬ 1 (i + 1) ² , and in the second

(13) |B ∩ (0; s ^m )|

s m ¬ b n

p i b n

¬ 1 i . Moreover, if k n

< j < k n

, then

(14) |B ∩ (0; s ^m )|

s m ¬ b n

a j+1 ¬ b n

a k

¬ 1

p ² _i+1 ¬ 1 (i + 1) ² .

From (5) and (12)-(14) it follows that 0 ∈ Φ ⁺ _hsi (B ⁰ ) = Φ ⁺ _f (B ⁰ ), contrary to (11). _■

Corollary 12 If T hbi ⊈ T hai then there is f ∈ A ¹ such that T hbi ⊂ T ^f ⊂ T hai and the topology T ^f is generated by no sequence.

References

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34 (2006), 37–47.

[2] M. Filipczak, T. Filipczak, Remarks on f-density and ψ-density, Tatra Mt. Math. Publ. 34 (2006), 141–149.

[3] M. Filipczak, T. Filipczak, On f-density topologies, Topology Appl. 155 (2008), 1980–1989.

[4] M. Filipczak, T. Filipczak, Density topologies generated by functions and by sequences, Tatra Mt. Math. Publ. 40 (2008), 103–115.

[5] M. Filipczak, T. Filipczak, J. Hejduk, On the comparison of the density type topology, Atti Sem. Mat. Fis. Univ. Modena 52 (2004), 37–46.

[6] M. Filipczak, J. Hejduk, On topologies associated with the Lebesgue measure, Tatra Mt. Math.

Publ. 28 (2004), 187–197.

[7] J.C.Oxtoby, Measure and Category, Springer-Verlag, New York Heidelberg Berlin, 1980.

[8] S. Scheinberg, Toplogies which generate a complete measure algebra, Adv. Math. 7 (1971), 231–239.

[9] W. Wilczyński, Density topologies, in: Handbook of Measure Theory, Elsevier Science B. V.,

Amsterdam, 2002, 675–702.

Małgorzata Filipczak

Faculty of Mathematics and Computer Sciences, Łódź University ul. Stefana Banacha 22, 90-238 Łódź, Poland

E-mail: malfil@math.uni.lodz.pl Tomasz Filipczak

Institute of Mathematics, Łódź Technical University ul. Wólczańska 215, 93-005, Łódź, Poland

E-mail: tfil@math.uni.lodz.pl

(Received: 20.06.2008)